The symmetry and simplicity of the

laws of physics and the Higgs boson

Juan Maldacena

Institute for Advanced Study, Princeton, NJ 08540, USA

Abstract

We describe the theoretical ideas, developed between the 1950s-1970s, which ledto the prediction of the Higgs boson, the particle that was discovered in 2012. Theforces of nature are based on symmetry principles. We explain the nature of thesesymmetries through an economic analogy. We also discuss the Higgs mechanism,which is necessary to avoid some of the naive consequences of these symmetries, andto explain various features of elementary particles.

1 A Fairy Tale

Our present understanding of particle physics is like the story of the Beauty and the Beast.Beauty represents the forces of nature: electromagnetism, the weak force, the strong forceand gravity. These are all based on a symmetry principle, called gauge symmetry. Inaddition, we also need the beast, which is the so-called Higgs field. It contains much of themysterious and strange (some would say “ugly”) aspects of particle physics. But we needboth to describe nature because we are the children of this marriage between the beautyand the beast.

Figure 1: Particle physics is like the Beauty and the Beast. The four fundamental forcesare Beauty and the Higgs field is the Beast.

Here we will attempt to introduce these two main actors: the beauty or gauge principle,and the beast or Higgs mechanism. We will do this through some analogies. There havebeen many attempts to explain particle physics to the general public. The usual expositionstake the reader through a long path starting from the world of ordinary experience, throughmolecules, atoms, nucleus, etc, usually following the order of historical discoveries. Herewe will attempt something different, we will just parachute into the modern description.We will find ourselves in a sort of fairyland, with very simple rules. Our main point isto emphasize the role of gauge symmetries and to highlight the surprising simplicity ofthese laws. These are the laws of the “Standard Model” of particle physics. They aresurprisingly simple given that they ultimately explain most of the ordinary phenomena.

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They describe the universe since it was a millisecond old till today. They also ultimatelydescribe almost all of known matter (dark matter is the exception). This description ofnature is the result of arduous experimental work, which we will not review here, but canbe found in many of the other popular descriptions of particle physics.

2 Symmetry

2.1 Ordinary symmetries

(a) (b)

(e)(f)

(d)(c)

(g)

Figure 2: Figures with various amounts of symmetry. (a) The square is symmetric underrotations by 90 degrees. (b) This rectangle is only symmetric under rotations by 180degrees. (c) The square is also symmetric under a reflection along the dotted line. (d)This figure is not symmetric under any reflection. However, it is symmetric under rotationsby 90 degrees. (e) This circle can be rotated by any angle. (f) This ring has the samesymmetries as the circle. (g) This figure has no symmetry.

First, let us say a few words about the notion of symmetry. Symmetry in physics isexactly the notion that we associate with symmetry in the colloquial language. It is atransformation that leaves an object unchanged. For example, a square can be rotatedby 90 degrees and it looks the same. If we had a rectangle, this would not be the case.See figure 2. The rectangle can be rotated by 180 degrees. We can also have figuresthat have no symmetry, see figure 2(c). A circle is very symmetric, it can be rotated byany angle and it remains the same. Two different figures can have the same symmetries.For example, a circle and a ring are different figures but they have exactly the samesymmetries. A situation could arise where we know what the symmetries of the objectare, but we might not know what the object really is. This is sometimes enough to makepredictions. For example, we can have a rigid object which is symmetric under rotations,

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with the symmetries of a circle. Then we know that it will roll smoothly on a table. It canbe a solid cylinder or a hollow cylinder, but both will roll smoothly on a table. Of course,in other respects the hollow and solid cylinder can behave differently. For example, onecan float in water and the other might not.

The symmetries we will talk about are a generalization of these more familiar ones.Their consequences will be to determine the forces of nature.

2.2 Electricity and magnetism reminder

Electric

FieldForce

Magnetic

FieldForce

Velocity

(a)(b) (c)

Magnetic

Field

Figure 3: (a) An electric field exerts a force on a charged particle along the direction of thefield. (b) A magnetic field exerts a force on a moving charged particle that is perpendicularboth to the magnetic field and the direction of motion. (c) The charged particle ends upmoving in a circle around the magnetic field.

Before starting our discussion, let us recall some facts about electromagnetism. Onepostulates the existence of electric and magnetic fields. We think of them as little arrowsat each point in spacetime. We feel the presence of these fields by their action on chargedparticles. Electric fields act on charged particles by pushing them along the direction of theelectric field. Magnetic fields only act on moving charges. In the presence of a magneticfield a moving charge feels a force that acts perpendicularly to the direction of its velocity.So if you were a charged particle and you are moving forwards in a vertical magneticfield, you would feel a force pushing you to your side. If there were no other forces, thenyou would end up moving along a circular trajectory. See figure 3. In summary, chargedparticles move in circles around magnetic fields.

These electric and magnetic fields have their own dynamics. We can have electromag-netic waves, which are interlocked oscillations of electric and magnetic fields. These canpropagate through the vacuum. They are radio waves, light, X-rays, gamma rays, etc. Allthis is hopefully familiar to you. Do not panic! You do not need to know the detailedequations to understand what follows. All you need to know is that there are electric andmagnetic fields, which can exist in otherwise empty space. These fields act on charged par-ticles and affect their motion. Electric fields push them along the direction of the electricfield. Charged particles move in circles around magnetic fields.

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2.3 Gauge symmetry

Electromagnetism can be viewed as a “gauge theory”. This is another point of view onelectromagnetism. This point of view is particularly useful for generalizing it to otherforces. It is also useful for describing the quantum mechanical version. To explain what agauge symmetry is, it is convenient to introduce an economic analogy∗. In this economicanalogy we will make a few idealizations and simplifications. Keep in mind that our goalis not to explain the real economy. Our goal is to explain the real physical world. Thegood news is that the model is much simpler than the real economy. This is why physicsis simpler than economics!

Bridge

Bank

1.5 dollars = 1 eurocountry dollar euro

peso

1 dollar = 10 pesos

Figure 4: Each circle is a country that has its own currency. They are connected byblue bridges. At these bridges there is a bank. When you cross the bridge, you shouldchange all your money to the new currency. We have indicated some of the currencies andexchange rates. At each bridge there is an independent exchange rate.

Now, let us get to the economic model. We imagine we have some countries. Eachcountry has its own currency. Let us imagine that the countries are arranged on a regulargrid on a flat world. See figure 4. Each country is connected with its neighbors witha bridge. At the bridge there is a bank. There you are required to change the moneyyou are carrying into the new currency, the currency of the country you are crossing into.

∗ The analogy between foreign exchange and lattice gauge theory was noted in K. Young, “Foreignexchange market as a lattice gauge theory”, American Journal of Physics 67, 862 (1999). Here we willextend that discussion, with the physics goal in mind.

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There is an independent bank at each bridge. There is no central authority coordinatingall the exchange rates between the various countries. Each bank is autonomous and setsthe exchange rate in an arbitrary way. The bank charges no commission. For example,assume that the currency in your original country is dollars and the one in the new countryis euros. Suppose that the exchange rate posted by the bank at the bridge between twocountries is 1.5 dollar = 1 euro. See figure 4. Then if you have 15 dollars the bank convertsit to 10 euros as you cross the border. If you decide to come back your 10 euros will beconverted to 15 dollars. Therefore, if you go to a neighboring country and you come rightback, you end up with your original amount of money. Another rule is that you can onlygo from one country to the neighboring country. From there you can continue to any ofits neighbors and so on. However, you cannot fly from one country to a distant countrywithout passing through the intermediate ones. You can only walk from one to the next,crossing the various bridges and changing your money to the various currencies of theintermediate countries. The final assumption is that the only thing you can carry fromone country to the next is money. You cannot carry gold, silver, or any other good. Wewill later relax this assumption, but for now we will consider this simplest situation.

Let us review the assumptions again. We have countries arranged in a grid. Eachcountry has its own currency. The countries are connected through bridges. There is abank at each bridge which exchanges money. The banks choose any exchange rate thatthey please. There is no commission. You can only carry money from one country to thenext. This is fairly simple. All you can do is travel between countries converting yourcurrency each time you cross a border.

Where is the symmetry? The gauge symmetry is the following. Imagine that one of thecountries has accumulated too many zeros in its currency and wants to drop them. This isfairly common in the real world in countries with high inflation. What happens is that oneday the local government decides that they will change their currency units. For example,instead of using Pesos now everybody needs to use “Australes”. The government declares1,000 Pesos will now be worth 1 Austral, or 1,000 Pesos = 1 Austral. So everybody changesall prices and exchange rates accordingly. If you needed to pay 5,000 Pesos for a banana,now you will need to pay 5 Australes. If your salary was 1 million Pesos, it will now be 1thousand Australes. Suppose the neighboring country is the USA. If the exchange rate was3,000 Pesos = 1 Dollar, it will now be 3 Australes = 1 Dollar. See figure 5. We call this a“symmetry” because after this change nothing really changes, nobody is richer or poorerand the change offers no new economic opportunities. It is done purely for convenience.You can see this gauge symmetry in action in some Argentinean banknotes in figure 6. Itis called a “gauge” symmetry because it is a symmetry of the units we use to measure or“gauge” the value of various quantities.

This symmetry is “local”, in the sense that each country can locally decide to performthis change, independently of what the neighboring countries decide to do. Some countriesmight like to do it more frequently than others. In the real world, Argentina has eliminatedthirteen zeros through various actions of this “gauge symmetry” since the 1960s, so that1 Peso of today = 1013 Pesos of the 1960s.

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Australes

3 Australes = 1 DollarDollar

banana =

5 Australes

3000 Pesos = 1 DollarPeso

banana =

5000 Pesos

Dollar

Currency

transformationGauge

transformation

1000 Pesos = 1 Austral

Figure 5: Each country can change its currency units. Here the country that was usingPesos changes the currency to Australes so that 1000 Pesos = 1 Austral. All prices andexchange rates change accordingly. Here we indicated how the price of a banana wouldchange. We have also indicated how the exchange rate to one of the neighboring countrieschanges. Of course, the exchange rates with all the neighbors change.

Now we will focus on speculators. A speculator is somebody who will travel alongvarious routes converting his money as he crosses each bridge. His goal is to earn money.He wants to travel along the paths with the highest monetary gain. Recall that accordingto our rules he has to actually travel to the different countries. He cannot sit at a deskand order his trades on a computer.

Do you think you can make money in this world? Think about it, what would you lookfor?

At first sight it seems that you cannot make any money. In fact, if you go from onecountry to its neighbor and back, you end up with the same amount of money. However,it is possible to make money if you come back a different way!

Just as an example let us consider three countries, say the USA, Europe and Argentina,with their corresponding currencies dollars, euros, and pesos. Now let us imagine that

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Figure 6: A gauge symmetry in action. Here we see a real world change from PesosArgentinos to Australes.

there are bridges connecting these three countries, see figure 7. The exchange rates are asfollows:

1.5 dollars = 1 euro , 1 dollar = 10 pesos , 1 euro = 10 pesos

In a situation like this, can you make money? Think about it before you continue reading.It is worth the effort!

We can earn money as follows. We start in Argentina with 10 pesos. We go to Europeand get 1 euro. We go to the USA and get 1.5 dollars. And then back to Argentina andwe get 15 pesos. We started with ten and ended with fifteen. The gain of this circuit isa factor of 1.5, or earnings of 50%. If you started the circuit with X pesos, you wouldhave 1.5 × X pesos at the end. This factor is independent of the currency units. If theArgentinean government changes from Pesos to Australes, then the gain of the circuit isstill the same, it is still a factor of 1.5.

You might think that the banks were dumb to set the “wrong” exchange rates andthat speculators are taking advantage of them. Well, this is according to the rules we haveenunciated. The banks set the exchange rates they want. With some choices there will beopportunities to speculate and with some others there will not be. The speculators have avery simple mind, they only care about earning money and they will choose the path thatmakes them earn most money. In the above situation the speculators would be moving incircles going from Argentina to Europe to the USA and back to Argentina. They wouldbe following the green line in figure 7.

Now, in physics the countries are analogous to points, or small regions, in space. Thewhole set of exchange rates is a configuration of the magnetic potentials throughout space.A situation like the one in figure 7, where you can earn money, is called a magnetic field.

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Argentina

Europe USA

1 Euro = 1.5 Dollar

1 Dollar = 10 Pesos1Euro = 10 Pesos

Euro

Peso

Dollar

Figure 7: Here we see three countries with the respective currencies. On the blue bridgeswe see the corresponding exchange rates. If you follow the circuit along the green line, youwould earn money. You would have a gain factor of 1.5 or earnings of 50% .

The amount of gain is related to the magnetic field. The speculators are called electronsor charged particles. In the presence of magnetic fields, they simply move in circles inorder to earn money. In fact, the total gain along the circuit is the flux of the magneticfield through the area enclosed by the circle. Now imagine that you are a speculator thathas debt instead of having money. In that case you would go around these countries inthe opposite direction! Then your debts would be reduced in the same proportion. In theexample of figure 7, your debts would be reduced by a factor of 1/1.5 by circulating in thedirection opposite to the green arrow. In physics, we have positrons, which are particles likethe electron but with the opposite charge. In fact, in a magnetic field positrons circulatein the opposite direction as compared to electrons.

In physics, we imagine that this story about countries and exchange rates is happeningat very, very short distances, much shorter than the ones we can measure today. When welook at any physical system, even empty space, we are looking at all these countries fromvery far away, so that they look like a continuum. See figure 8. When an electron is movingin the vacuum, it is seamlessly moving from a point in spacetime to the next. In the verymicroscopic description, it would be constantly changing between the different countries,changing the money it is carrying, and becoming “richer” in the process. In physics wedo not know whether there is an underlying discrete structure like the countries we havedescribed. However, when we do computations in gauge theories we often assume a discretestructure like this one and then take the continuum limit when all the countries are very

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(a)(b) (c)

Zoom out Zoom out

Figure 8: Here we display the grid of countries at various scales. We are zooming outas we move to the right. If we zoom out sufficiently we can view the whole grid as acontinuum.

close to each other.Electromagnetism is based on a similar gauge symmetry. In fact, at each point in

spacetime the symmetry corresponds to the symmetry of rotations of a circle. One wayto picture it is to imagine that at each point in spacetime we have an extra circle, anextra dimension. See figure 9(a). The “country” that is located at each point in spacetimechooses a way to define angles on this extra circle in an independent way. More precisely,each “country” chooses a point on the circle that they call “zero angle” and then describethe position of any other point in terms of the angle relative to this point. This is likechoosing the currency in the economic example. Now, in physics, we do not know whetherthis circle is real. We do not know if indeed there is an extra dimension. All we knowis that the symmetry is similar to the symmetry we would have if there was an extradimension. In physics we like to make as few assumptions as possible. An extra dimensionis not a necessary assumption, only the symmetry is. Also the only relevant quantitiesare the magnetic potentials which tell us how the position of a particle in the extra circlechanges as we go from one point in spacetime to its neighbor.

In electromagnetism the electric and magnetic fields obey some equations, the so-calledMaxwell equations. In the economic analogy this would be analogous to a requirement onthe exchange rates. In the economic model we can intuitively understand this requirementas follows. Let us imagine we have a configuration with generic exchange rates. Speculatorsstart carrying money around. Suppose we focus on a particular bridge, where a particularbank sits. There will be speculators crossing this bridge in both directions. However, ifthere are more speculators going in one direction than in the other direction, then thebank might run out of one of the currencies. For example, consider the bank sitting at abridge that connects Pesos to dollars. If there are more speculators wanting to buy dollarsthan there are speculators wanting to buy pesos, the bank will run out of dollars. If thishappened in the real world, then the bank would adjust the exchange rate so that therewould be fewer speculators wanting to buy dollars. In fact, if we assume that the numberof speculators following a particular circuit is proportional to the gain that they will have

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along this circle, then one finds that the condition for banks not to run out of either ofthe currencies, or that the net flow of money across each bridge is zero, is equivalent toMaxwell’s equations. The mathematically inclined reader can find the derivation in theappendix.

(a) (b)

Electromagnetic Weak

Figure 9: (a) The electromagnetic interaction has the same symmetries as a configurationwhere we have a circle at each point of spacetime. Here each spacetime point is wherethe black lines intersect. We can think of the circle as an extra dimension. (b) The weakforce has the same symmetries as a configuration where we have a sphere at each point inspacetime. We do not know whether the circles or spheres really exist as extra dimensions.What we do know is that the gauge symmetry is the same as if they existed. The circlesof spheres are useful for visualization but we only think about their symmetry and focusonly on the associated “exchange rates”.

2.4 The weak force

Let us now turn to the weak force. It is the force responsible for radioactive decays. Forexample, a free neutron (outside a nucleus) decays in about 15 minutes to a proton, anelectron and a neutrino. This is a very slow decay compared to other processes that happenon microscopic time scales. The weak force is not terribly relevant for our everyday life.However, despite its weakness, it played an important role in the history of the universe.More specifically, in the synthesis of the chemical elements in stars. In fact, all the chemicalelements around us, except for hydrogen and helium, were “cooked” in stars. The weakforce played a crucial role in this process. Closer to home, we can say that the weak forcecan move mountains! More precisely, weak decays inside the earth are partly responsiblefor maintaining the earth hot, which in turn moves the continents, creating the mountains!

The weak force can also be understood using a gauge theory. In this case, at eachpoint in space we have the symmetries of a sphere, let us call it the weak sphere. See

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figure 9(b). We do not know whether the sphere is real or not. What we do know isthat when we go from one point in space to another we have to specify three quantities,three “exchange rates”. Instead of carrying money, we are now carrying an object that hassome orientation in the weak sphere. If we start at a country with an object in the weaksphere, as we go to the neighboring country we have to re-orient the object according tothe “weak exchange” rates. We need three quantities since we have to specify a rotationaxis (two quantities) and an angle of rotation around that axis (the third quantity). So,instead of one magnetic field, we have three different types of magnetic fields. There areequations, similar to the ones for electromagnetism, which govern the behavior of thesemagnetic fields together with the corresponding electric ones. These equations were firstproposed by Yang and Mills in 1954. When W. Pauli heard about it, he strongly objected.Pauli said that the Yang-Mills theory implied that there would be new massless particles,which are not observed in nature. This was a beautiful theory killed by an ugly fact.

1 1 1 1 1 1

1 1 1 1 1 1

1.1 1.3 11 1.2 1.2 1.1

1 1 1 1 1 1

1 1 1 1 1 1

1=1 1=1.3 1=1.3 1=1.31=1 1=11=1

short wavelength

gain = 1.3

Long wavelength

gain = 1.1

Figure 10: We can see a configuration of exchange rates with a long wavelength or a shortwavelength. The wave consists in the fact that the numbers in the middle go up and downas we move from left to right. Each segment is a bank sitting between two countries. Thecountries sit at the line intersections. The number indicates the exchange rate when youcross the bridge in the direction of the arrow. Notice that the total amplitude of the waveis the same; the exchange rates go between 1 and 1.3. The gain obtained by following anelementary square circle, indicated by the green lines, is smaller for the longer wavelengthconfiguration.

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2.5 Why massless particles?

To understand Pauli’s objection, let us first focus on some properties of waves. In thesystems we consider we can have waves with different wavelengths. The wavelength ofa wave is the distance between two successive maxima. In physical systems one is ofteninterested in the energy cost for exciting waves of various wavelengths. For a wave with agiven amplitude this energy cost can depend on the wavelength. For an electromagneticwave, this energy cost decreases when we make the wave longer and longer. It turns outthat the mass of the particle is related to the energy cost to excite a very long wavelengthwave. This is related to the famous formula E = mc2. Unfortunately I have not found ashort way to explain this, so you will have to trust me on this. In our economic analogywe have not talked about energy. Let us simply say that the energy increases as the gainavailable to speculators increases. This makes intuitive sense, the more the speculators canearn, the harder it is for the banks! Therefore configurations with less gain have a lowerenergy cost. In figure 10 we describe a whole sequence of exchange rates with a long and ashort wavelength configuration. The crucial point is that the gain that speculators obtainby following the elementary square circuits (denoted by the green arrow in figure 10) is onlyrelated to the difference between neighboring exchange rates, but not on their absolutemagnitude. Therefore, the longer the wavelength, the smaller the difference. The fact thatthe gain becomes smaller as the wavelength becomes longer implies that the associatedparticle, the photon, is massless. This is a correct argument for electromagnetism and it isalso correct for the weak force for the same basic reason. At least it is true for the versionof the weak force described so far...

3 The Higgs mechanism

Now, it turns out that there is a way around this argument. This is called the Higgsmechanism, which was proposed by many researchers†. Here we will explain it usingthe economic analogy. So far, we have assumed that you can only carry money betweencountries. Now, let us assume that you are also allowed to carry gold. So the new ruleis that you are allowed to carry gold and/or money between different countries. Gold hasa price in each country, which is set by the inhabitants of each country independently ofthe others. A savvy speculator realizes that a new opportunity opens up. You can nowbuy gold in one country, take it to the next, sell it, and bring back the money to the firstcountry. As an example, say that the exchange rate between pesos and dollars is 4 pesos= 1 dollar. And the price of gold in Argentina is 40 pesos per ounce and the price in theUSA is 5 dollars per ounce. What would you do? Again, the prices and exchange ratesare

4 pesos = 1 dollar , 1 ounce = 40 pesos , 1 ounce = 5 dollars

† These include Anderson, Brout, Englert, Goldstone, Guralnik, Hagen, Higgs, Kibble, Nambu, etc.The detailed history can be found elsewhere.

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Think about it, do not continue reading until you have the answer. It is a bit hard, butworth the effort! Yes, indeed, you would start with five dollars in the USA, buy gold there,go to Argentina, sell it for 40 pesos, go back to the USA and get 10 dollars when you crossthe bridge back. This operation then has a gain of a factor of two, or a 100% profit.See figure 11. Note that we continue to have the gauge symmetry. If the Argentineangovernment changes the currency to Australes, your gain would be the same. Now, we canuse this gauge symmetry to choose the currency so that the price of gold is the same in allcountries. Let us call the new units new pesos and new dollars. Now the price of gold is 1New peso per ounce and 1 New dollar per ounce. However, the exchange rates might notbe one to one. In fact, they cannot be one to one if originally there was an opportunityto speculate. For the example in figure 11 the new exchange rate is 1 New peso = 2 Newdollars. Note that this is not a “gold standard” that removes all exchange rates. It is veryimportant that the exchange rates are still present.

In summary, now the new prices and exchange rates are

1 new peso = 2 new dollars , 1 ounce = 1 new peso , 1 ounce = 1 new dollars

If you are a speculator, now it is easier to see what to do, isn’t it? With these new currencyunits obtained by setting the price of gold to one, one can immediately see that if anyexchange rate is different than one to one, then there is an opportunity to speculate byperforming the gold circuit. The opportunity to speculate remains, and the gain remainsa factor of two, or 100% profit. As usual, the net gain does not change when we changecurrency units.

USAArgentina

4 Pesos = 1 Dollar1 ounce = 1 ounce = 5 Dollars40 Pesos

Pesos Dollars

Gold

money

gain = 2

Figure 11: Here we see two countries with their respective gold prices. We have also giventhe exchange rate between them. For these prices and exchange rate it is possible to earnmoney by buying gold in the USA, taking it to Argentina, selling it there and bringingback the money, exchanging it at the bank, of course. The net gain is a factor of two or100%. Gold is indicated in yellow and money in green.

Now, a crucial new feature of this new economic model is that the gain does notbecome smaller and smaller as we increase the wavelength. The reason is that now youcheck whether you can gain money by looking at a single bridge, and you do not need

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to compare neighboring bridges. Once we set the price of gold to one everywhere, thenany exchange rate that is different from one to one leads to an opportunity to speculate.In the physics version, this means that it now costs some energy to move any exchangerate away from one to one. This cost is present even for long wavelength configurations.In physics, this leads to a massive particle, to a massive photon. A mechanism of thissort is happening inside superconductors, and this was indeed an inspiration for the Higgsmechanism.

In physics, we think that a similar mechanism gives rise to a mass to the particles thatmediate the weak force. Instead of gold, we now have some object at each country, orspacetime point, that has some orientation in the weak sphere. This is the so-called Higgsfield. We can use this object to completely fix the orientation of each weak sphere. Nowthe object is oriented in the same way everywhere. This is analogous to fixing the price ofgold to one. We still have the weak “exchange rates” which tell us how we get rotated inthe weak sphere when we go from one country to the next. The zero gain configuration iswhen we are not rotated at all. In that case we can say that the weak exchange rates areall “one to one”. If we are rotated in any way, there will be a possibility to speculate. Asa consequence the weak force mediators are massive. In this way, the heroic Higgs fieldhas rescued the beautiful lady. We can explain the weak force using gauge symmetry and,at the same time, avoid having massless particles.

3.1 Apologizing for one oversimplification

We have made an implicit oversimplification in this description, which can confuse a veryattentive reader. We have given the impression that the theory of electromagnetism andthe weak force arise from a gauge theory with the symmetries of a circle and a sphererespectively. In nature, the W± bosons are charged. A charged object is one that ischanged under a gauge transformation. This would be impossible if electromagnetismcorresponded to a circle completely independent of the weak sphere. In fact, while itis correct that we start with the symmetries of a circle and a sphere, the symmetry ofelectromagnetism corresponds to a combination of a rotation on the circle, as well as arotation on the sphere. Then the W± bosons are charged because when we perform anelectromagnetic “gauge transformation” we are also performing a rotation on the weaksphere and it is therefore not surprising that we can change the W± bosons. Similarly, wewill see below that the electron and the neutrino represent the same particle but rotatingin different ways in the weak sphere. However, the electron and neutrino have differentcharges. The neutrino, as it name indicates, is neutral or uncharged, while the electronis charged. But, since electromagnetism includes a rotation on the weak sphere, thisdifference in rotation in the weak sphere translates into a different charge. For this reasonthe whole theory is called electroweak theory.

This is an important point to get the details right, but we will ignore it in the rest ofthe discussion.

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4 Quantum mechanics

The system we have described so far, through the economic analogy, gives rise to what isnormally called a classical field theory. A field is a quantity that is defined at each pointin spacetime. For example, the price of gold is a field; at each point in spacetime it takesa definite value. Similarly, the exchange rates are also fields. For each point, we have oneexchange rate per spacetime dimension, since the number of neighbors that a country hasis proportional to the dimension of spacetime.

In physics this is not the whole story. The dynamics of these fields is ruled by the lawsof quantum mechanics. An important feature for these laws is that they are probabilistic.One might think that in the vacuum all the fields are zero. However, this is not the case,they take random values. All we can say is that they are given by a certain probabilitydistribution. In the economic analogy, we can say that the values of the exchange ratesand the price of gold, are all random. This randomness follows a very precise law, whichis encoded in the precise form of the probability distribution. We will not give the preciseformula here (it can be found in the appendix), but we simply note that it is such thatconfigurations of exchange rates and gold prices become less probable when there aregreater opportunities to speculate. We have a precise law for the probability of eachconfiguration, but we cannot predict with certainty, which of the possible configurationswill occur when we look at the system.

Note that all these fluctuations happen at short distances. If we follow a very bigcircuit, then we pass through many countries and their exchange rates all average out.In the vacuum, at long distance they average out to zero so that we recover the classicalresult where the fields are all zero.

The probability cost that we have to pay when we set the exchange rates to valuesleading to larger speculative opportunities is also related to the energy cost we discussedabove. They are essentially the same. Higher energy configurations are less probable. Innature, the particles that carry the weak force are very massive. They weigh around ahundred times the mass of the proton, which is a lot for an elementary particle. They arecalled the W+, W− and Z bosons. Their large mass explains the weakness of the weakforce. This large mass implies that we are very unlikely to produce fluctuations in the“weak exchange rates”. Therefore, a particle that interacts only through the weak force,such as the neutrino, is very difficult to see. In fact, a few per cent of the energy of thesun comes out in neutrinos‡. However, we are totally oblivious to these neutrinos. Theysimply pass through us day and night and we do not see them. You need very big detectorswith very sensitive electronics to catch a very tiny fraction of them.

4.1 The continuum limit and the Higgs boson

The mechanism described above does give a mass to the mediators of the weak force,but it does not explain why there should be a new physical particle, such as the Higgs

‡This shows that the weak force is important for the workings of the sun.

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boson. Let us explain this through the economic analogy. We can fix the price of goldto one everywhere by a currency (or gauge) transformation. Once this is done, the onlyremaining variables are the exchange rates. In physics this gives rise to a massive particle(with spin one), but to no other particle. In classical physics, this theory is perfectlyconsistent without an extra particle. However, in the quantum mechanical version this isnot the case, especially in the case of the weak force.

The quantum mechanical version the continuum limit, indicated graphically in figure8, is very subtle. A detailed analysis reveals that a theory without extra fields would notpermit the weak force mediations to have a mass which stays fixed as we take the spacingbetween the countries to zero. Their mass would go to infinity as we take the continuumlimit in the quantum theory. For this reason everybody expected that the Large HadronCollider would discover new particles. The simplest possibility was to add just one newparticle.

In the economic analogy these new particles arise when we have more goods that wecan carry between the different countries. For example we can also have silver. Silver alsohas a price in each country, again chosen arbitrarily by the inhabitants of each country.Now, in each country, the ratio between the price of gold and silver is independent of thecurrency units, it is gauge invariant. If one ounce of gold costs 2,000 pesos and one ounceof silver costs 1,000 pesos, then we can say that gold is twice more expensive than silver.If we change currencies to Australes, as in figure 5, the price of gold is 2 Australes andthat of silver is 1 Austral. But gold is still twice more expensive than silver. Therefore, inthis situation, we have a quantity that is defined in each country which is independent ofthe choice of currency. In physics this corresponds to a field that gives rise to a physicalparticle. This is the field associated to the actual physical Higgs boson particle. Note thatin the case that we only had gold, we also have a quantity defined in each country, which isthe price of gold. However, this quantity depends on the choice of currency. Quantities thatdepend on the currency units do not reflect real economic opportunities. In physics theyare not observable. In fact, we have seen that through a gauge transformation (or currencytransformation) we can set the price of gold to one everywhere, so that it disappears as aphysical field. However, with both gold and silver we have now a physical field.

One extra field is the simplest possibility. It predicts one new particle. Indeed thisextra particle was discovered at the Large Hadron Collider in 2012.

Though we have not attempted to give the history of these concepts, we cannot resistmaking a few historical comments. Yang and Mills invented the Yang Mills theory todescribe mesons. However, we currently use it to describe weak and strong interactions.This is an example where a good idea was not useful for its original purpose but was usefulfor other problems. Though Pauli’s objection was right, it was not a fatal flaw. It could befixed in a relatively simple way. The mathematics of gauge theories was discovered beforeby mathematicians who know these structures as fiber bundles. Inspired by many previousexperiments and partial results, the theory of electro-weak interactions was written firstby S. Weinberg, with important contributions by Glashow and Salam. Later experimentsconfirmed this theory and ruled out competing alternatives. The most important current

16

experiment in particle physics is going on at the Large Hadron Collider in Europe.Notice the simplicity of physics relative to economics. In economics there are many

things that we can trade, not just gold and silver. We can also trade among all the countriesat the same time. In physics we can trade only with the neighbors. It is worth noting thatWeinberg’s paper is just three pages long, while here it has taken us many more pages togive a still imprecise explanation. This is an example of the power of formulas. It is oftensaid that a picture is worth a thousand words. A formula is worth a million pictures!

Modern

Weak

Figure 12: On the left we see Atlas holding the celestial spheres. On the right we see thepattern of spheres whose symmetries determine the weak force. The ancients had a fewspheres per planet. In some sense, we now have one sphere per spacetime point.

One of the detectors of the LHC is named Atlas. In figure (12) we can see the mythicalAtlas holding the celestial spheres. Probably when you look at this picture you think: Oh,how naive these Greeks were, with all those spheres. How much simpler is the Newtoniandescription. Now, the modern view of nature has the symmetries of a sphere at each pointof spacetime! So the number of spheres has grown a great deal. However, the structureis governed by a rather simple symmetry. The weak sphere is very simple. The modernAtlas detector is reaching into the weak sphere, uncovering its secrets§.

5 Massses for other particles

It is often said that the Higgs field gives a mass to all other particles. In fact, from whatwe said so far, we could add masses for the other matter particles, such as the electron,with or without the Higgs. The real reason we need the Higgs to give mass to the electron

§ My apologies to the other LHC detectors which has less mythical names. From the scientific pointof view, they are as important as Atlas!

17

is related to a strange property of the weak interactions. To explain this weird feature, weneed to describe in more detail some of the properties of elementary particles. We need totake into account that the electron has spin. First we will describe spin, and then describethe weird feature of the weak interactions.

Figure 13: On the left we see a left handed electromagnetic wave. The arrows point alongthe direction of the electric field at one instant in time. On the right side we see a righthanded electromagnetic wave. Now the electric field rotates in the opposite direction asthe wave propagates. Both waves propagate to the right.

You might be familiar with the fact that light can be polarized and that it has twopolarization states. For example, polarized sunglasses are designed to block the predom-inantly polarized light produced by reflection glare. In fact, it is convenient to think interms of circularly polarized light. In figure 13 we see a left and a right handed electromag-netic wave. These waves carry angular momentum around the direction of propagation.In fact, the photons that form these waves also carry angular momentum. In the economicmodel, the fact that light is polarized is related, to the fact that the exchange rate connectsa country to its neighbors. These neighbors can be in any of the directions transverse tothe wave. In figure 10 we had a wave going to the right and the exchange rates that werechanging where the ones going up, a direction transverse to the wave.

Electrons and neutrinos also share this property with light. They can also be polarized.They also carry angular momentum, or some amount of rotation. When a particle ismoving we have a preferential direction to define the angular momentum. This angularmomentum can be left or right handed as in the case of light. If the particle is at rest wedo not have a preferential direction. The angular momentum can point in any direction.All directions are related by the rotation symmetry of empty space.

Something special happens for massless particles. A massless particle is always moving.If the angular momentum is along the direction of motion, then in all reference frames itwill be along the direction of motion. This is not the case for a massive particle. Imaginea massive particle moving up in space, with the angular momentum along the directionof motion. Let us now decide that we are also moving along the vertical direction, fasterthan the particle. In our moving reference frame the particle looks like it is moving down.But its spin is unchanged. So the direction of the spin is now opposite to the directionof motion. In other words, by going to a moving reference frame, we have changed the

18

direction of the velocity relative to the direction of spin. The conclusion is that for amassless particle the notion of whether the spin points along the direction of motion oropposite to it is some characteristic of the particle, independent of the reference frame.Recall that the laws of physics should be the same independent of how we are moving.This is the principle of relativity.

With these preliminaries, we can now turn our attention to the weird property of theweak interactions. The weak interactions treat differently left and right handed particles.The weak interactions affect only left handed particles. This is a very surprising property,which is possible only because the weak interactions violate reflection symmetry. What isreflection symmetry? This is the symmetry under reflections on a mirror. Imagine thatwe look at the world through a mirror. Is the world we see consistent with the laws ofphysics? From our everyday experience we would naively think that it should be. It israther hard to realize that you are looking at the world through a mirror. If you look atwritten text, then you would know, but this is only because we all use the same conventionto write. The question is whether the fundamental laws are the same or not. An importantproperty of reflections in a mirror is that the reflection of a rotating particle appears torotate in the opposite direction. See figure 14. Therefore, if the weak interactions treat leftand right handed particles differently, then the weak interactions can distinguish betweenthe real world and its reflection in a mirror. We have emphasized that the laws of physicsare based on interesting new symmetries, but here we encounter a very simple candidatesymmetry that the fundamental laws do not have! The laws of nature have some unfamiliarsymmetries but they lack a very familiar one.

Figure 14: If the tangerine is rotating in the direction of the arrow, then its reflectionwould rotate in the direction of the arrow in the reflection. Therefore, if the original rotatesin one direction, the image rotates in the opposite direction. A tangerine can indeed rotatein any of these two directions. But if we replaced the tangerine by a neutrino moving inthe vertical direction, then its mirror image would not look like an allowed particle.

A left handed electron and the left handed neutrino are basically the same particlebut moving in different ways on the weak sphere. The weak force transforms one into theother. The right handed electron does not feel the weak force. We do not know whethera right handed neutrino exists or not, it has not been detected yet, and it does not haveto exist. This sharp distinction between left and right handed particles is possible if the

19

particles are moving at the speed of light, so that their handedness is an intrinsic property.However, the electron is a massive particle. This is possible only due to its interaction

with the Higgs field. This is a new interaction that we need to postulate by hand, in orderto get the theory to agree with experiment. Through this interaction, an electron which ismoving at less than the speed of light, can be viewed as a particle with an identity crisis.Part of the time it is a left handed electron and part is a right handed electron moving inthe opposite direction. In average, it moves at less than the speed of light. The interactionwith the Higgs field turns one into the other. See figure 15

Space

Time

lefthanded

lefthanded

Space

Timespeed of light

speed of light

righthanded

(b)(a)

velocity

Figure 15: In (a) we see left and right handed massless electrons. The black arrow isthe direction of the spin of the electron. Whether it is left or right handed depends onthe direction of the spin relative to the direction of motion. So the two blue lines describeleft handed electrons since the direction of spin is opposite to the direction of motion. In(b) we see a massive electron. Some of the time it behaves as a right handed electron andsometimes as a left handed electron. It switches from one to the other thanks to the Higgsfield.

The quarks, which are the particles inside the proton or neutron, also get a massthrough this mechanism. There are further elementary particles similar to the electron,neutrino and quarks, which are more massive. These are unstable and decay quickly viathe weak interactions. All of these get masses through the interaction with the Higgs field.These interactions are all introduced by hand in order to fit nature. They do not followfrom any symmetry principle. Their values range over many orders of magnitude. Forexample, the top quark is about three hundred thousand times heavier than the electron.The neutrinos also get a mass in a similar fashion, though the details are a bit morecomplicated and have not been completely settled yet.

Despite the fact that the Higgs gives mass to most elementary particles, most of themass of ordinary objects does not come from the Higgs field! In fact, most of the masscomes from the mass of the protons and neutrons. These are composite objects. They

20

contain quarks which are moving very rapidly and most of the mass comes from the energyof this rapid motion. (Recall E = mc2.) But the Higgs is important for an importantmacroscopic property of the real world. The Higgs field sets is the size of atoms. The sizeof atoms is inversely proportional to the mass of the electron. The lighter the electron thebigger the atom. By changing the magnitude of the Higgs field it is possible to make allelementary particles lighter. For a person trying to loose weight, it would not be a goodidea to try to change the magnitude of the Higgs field. Ignoring the fact that this wouldbe exceedingly difficult to do, it would not have the desired effect. The mass would bealmost unchanged but the person’s size would become much bigger!

Given that the Higgs does all these good things for us, why do we say that the Higgsis ugly? One reason is that it represents a new force of nature that is not based ona gauge symmetry. A more practical one is that most of the unpredictable parametersof the Standard Model are associated with the interactions with the Higgs field. Theseparameters range over many orders of magnitude. By comparison the strength of the threegauge theory forces are somewhat similar, at high energies. Finally, the strangest featureof all is the value that sets the scale of the Higgs mass, and as a consequence the overallvalues of all the other masses. It is unclear what physics gives rise to this mass scale, and itis a mass scale that is much lower than the other fundamental mass scale in nature whichis the mass scale of gravity, which sets the strength of gravity. This can be understood asfollows. In the economic model, we have said that we recover the continuum description bysaying that the distance between the countries is much smaller than the shortest distanceswe can measure. In a world without gravity this distance could be infinitesimally small.In our universe we have gravity. The Einstein theory of gravity says that spacetime isdynamical. We also expect it to be quantum mechanical. Quantum mechanics says thatspacetime itself is fluctuating. In the economic model, spacetime is the grid of countries.Having the grid fluctuate means that countries can exchange neighbors, new countriescould appear and disappear, etc. In nature, all of this would be happening at a very shortdistance, a distance set by the strength of gravity. This turns out to be an exceedinglysmall distance. A distance 1016 times smaller than the shortest distance we can see todaywith our most powerful microscope, which is the Large Hadron Collider. The problem withthe mass of the Higgs particle is this question of why are the weak interaction phenomenahappening at a distance scale so much larger than this basic gravitational scale? We donot know.

There are several ideas for understanding these strange features of the Standard Model.Many of these ideas postulate the existence of new particles. Perhaps, they will be soondiscovered at the Large Hadron Collider. We are waiting expectantly.

In this exposition we did not talk about the strong interactions. It is the force thatholds the quarks together in the proton or neutron. It is also based on a gauge interaction.To answer Pauli’s objection, it uses a another mechanism which is inherently quantummechanical.

We should end by mentioning that we have very strong evidence for the existence of anew particle. This evidence comes from astronomical observations which see more matter

21

than what can be accounted for with the particles we already know. This is the so-called“dark matter”. A good candidate for a dark matter particle is a new particle that is subjectto the weak interactions. A so-called WIMP (Weakly Interacting Massive Particle). It isa good candidate because the cosmological abundance of these particles would be in theright range to fit observations. It is also predicted naturally by theories that try to explainsome of the puzzles of the Standard Model. It is also possible that dark matter is notrelated at all to the weak interactions.

Hopefully, once we have a more complete understanding we will find that the Higgsfield, which we view as an ugly component of the Standard Model, will be transformedinto a handsome prince. Or at least it will be part of the handsome prince. And they willlive happily ever after... or at least till the universe decays.

T HE ENDAcknowledgmentsI thank Graham Farmelo for encouraging me to write this up. I thank N. Arkani Hamed

and C. Morgavio for comments on the draft. I thank Wikipedia for providing some of thepictures.

I was funded in part by DOE grant de-sc0009988.A video presentation based on this article can be found in:https://www.youtube.com/watch?v=OQF7kkWjVWM

6 Mathematical appendix: A more quantitative de-

scription of the economic analogy

Warning: This appendix is designed only for people with the right mathematical back-ground.

Imagine that we have “countries” arranged on a d dimensional lattice labeled by points~n = (n1, n2, · · · , nd), where each ni is an integer number. For our spacetime we would taked = 4. Each point in the lattice is a country and is labeled by the vector ~n. Let us considerthe country sitting at the point ~n and its neighbor in the ith direction, sitting at ~n + ~ei,with ~ei = (0, · · · , 0, 1, 0 · · · 0) where the 1 is at the ith place. Here i ranges from one to d.The exchange rate between these two countries can be written as

R~n,i = eAi(~n)

Here R~n,i is the exchange rate and Aj(n) is simply its logarithm, which we introduce forlater convenience. If the country at point ~n uses Pesos and the country at point ~n + ~ejuses Dollars, then R~n,j tells us how many Dollars you get for one Peso.

Now a gauge transformation at point ~n changes the local currency by multiplying itby a factor f(n), which we write as

f(~n) = eε(~n)

22

This changes all the exchange rates connected to this point. More explicitly, under arbi-trary currency unit transformations the exchange rates change as

R~n,i →1

f~nf~n+~eiR~n,i , or Ai(~n)→ Ai(~n) + ε(~n+ ~ei)− ε(~n) (6.1)

A (n)

A (n) A (n + e )

A (n + e )

i

ij

ji

j

i

j

n+ ej

n n+ ei

n+ e + ei j

Figure 16: Elementary monetary speculative circuit. We start with some money at thecountry at position ~n. First we move to its neighbor in the position ~n+~ei. Then we moveto another neighbor at ~n + ~ei + ~ej. Then to its neighbor at ~n + ~ej. Finally we return tothe original country. Following this circuit, and carrying only money, we can earn a profitgiven by the magnetic flux, (6.2).

When we go though an elementary basic square circuit, see figure 16, the gain factoris given by

gain = R~n,iR~n+~ei,j1

R~n+~ej ,i

1

R~n,j= eFij(~n)

Fij(~n) = Aj(~n+ ~ei)− Aj(~n)− [Ai(~n+ ~ej)− Ai(~n)] (6.2)

where we defined the “magnetic flux” Fij for the corresponding elementary square circuit.Note that the gain factor, or the magnetic flux, is invariant under the change of currency,or gauge transformation, given in equation (6.1). When the gain factor is less than oneyou are losing money.

23

p(n+ e )i

Gold

A (n)i

p(n)

n n+ ei

Figure 17: Elementary speculative gold circuit. We start from the country at position ~nand buy Gold. We take it to the neighboring country at ~n+~ej. We sell it there. We bringback the money to the original country. Here Gold is yellow and money is green.

We now consider the same but with the addition of gold. Let us write the price of goldas

p(~n) = eϕ(~n) (6.3)

Now there is a new opportunity to speculate by taking gold and bringing back money, seefigure 17. The gain is

gain =p(~n+ ~ei)

p(~n)R~n,i= eDiϕ(n) , Diϕ(n) ≡ ϕ(~n+ ~ei)− ϕ(~n)− Ai(~n) (6.4)

where we have defined Diϕ(n), which in physics is called the “gauge invariant gradient ofthe field ϕ”. We will call this the “gold gradient”. It parametrizes the effective gain of thegold circuit, just as the magnetic flux (6.2) was parametrizing the gain of the monetarycircuit.

6.1 Quantum mechanical version

We can define a probabilistic version of the above model by assuming that the exchangerates are random. They are random variables drawn from a probability distribution thatdepends on the magnetic fluxes. We further assume that the probability distribution hasa local form, so that the probabilities of separated circuits simply multiply as if they wereindependent events. The price of gold is also a random variable. There is a probability forthe money circuit and also for the gold circuit, see figures 16, 17.

More concretely, the probability for a given set of exchange rates and gold prices is

P [A,ϕ] =∏~n,i,j

µ(Fij(~n))∏n,i

µ̃(Diϕ(n)) (6.5)

The product runs over all countries, which are labeled by ~n. For each country we alsomultiply over all the elementary money and gold circuits that pass through that country.Here µ(Y ) and µ̃(Y ) are functions which are both peaked at zero. This gives the highestprobability to the case where we have no opportunity to speculate. We will further assumethat these probabilities can be written as

µ(Y ) ∼ e−Y2+... , µ̃(Y ) = e−σ

2Y 2+···

24

where the dots represent higher order terms will not be important in the continuum limit.Here σ is just a parameter. In this situation the probability distribution (6.5) simplifiesto

P [A,ϕ] = e−∑

n[∑

i,j Fij(~n)2+σ2

∑i(Diϕ(~n))

2] (6.6)

In physics this gives the actual probability for finding the corresponding magneticpotentials in the vacuum through the following procedure. We consider a four dimensionalEuclidean lattice of this form. We focus on the particular set of exchange rates and goldprices sitting at n4 = 0. This is a particular three dimensional sublattice of the originallattice of countries. This surface can be interpreted as a discrete version of physical spaceat some instant of time, say at t = 0. The probabilities for the links at n4 = 0 in the aboveEuclidean model are essentially the same as the full quantum mechanical probabilities formeasuring the corresponding values of the magnetic potentials at an instant of time, sayt = 0, in the vacuum.

We say “essentially” because we still need to take the continuum limit, indicated infigure 8. This can be done as follows. We imagine that each point in space is given by~x = a~n where a is a very small number that goes to zero and ~n goes to infinity so that~x stays fixed. In this situation it is convenient to introduce a new magnetic potential ,Ai(~x), that will stay fixed as we take the continuum limit

Ai(~n)→ aAi(~x) , ϕ(~n)→ φ(~x)

We also have

Fij(~n)→ a2Fij(x) , Fij(~x) ≡ ∂Aj∂xi− ∂Ai∂xj

Diϕ(~n)→ aDiφ(x) = a

[∂φ(~x)

∂xi−Ai(~x)

]P [A, ϕ] = e−

∫dxd[

∑i,j F2

ij+m2∑

i(Diφ)2] m2 ≡ σ2

a2(6.7)

Now, this is the full story with a massive spin one field. This is the real physicaltheory. As you see it is not that complicated. We have obtained it from a relatively simpleprobabilistic economic model. What is somewhat complicated is to relate this descriptionto actual physical measurements. The final parameter m is the mass of the spin oneparticle. We recover ordinary electromagnetism by setting m = 0.

As we have said we can get the quantum mechanical probabilities for the vacuum from(6.7) by fixing the Ai(x) fields at one instant of Euclidean time xd = 0 and then integratingout over the fields at all other times. The description of quantum mechanical processesinvolving time, such as observations at different values of ordinary time, is more compli-cated and would require a longer discussion of quantum mechanics. But conceptually, it isbasically the same as what we have done so far. The rest of the forces of particle physics,the weak force and the strong force can be incorporated through a similar description.Some details are somewhat complicated due to the need to incorporate variables that areanticommuting “numbers” to describe electrons, neutrinos or quarks.

25

Finally, Maxwell’s equations can be simply derived from the condition that the ex-change rate variables Ai(x) are such that they minimize the probability. Even thoughthe fundamental description is random we can try to find the average exchange rates andgold prices that maximizes the probability (6.6) or (6.7). These are obtained by tak-ing the derivative of the exponent in (6.6) with respect to Ai(~n) and ϕ(~n) and settingthese derivatives to zero. This gives a discrete version of the classical equations of motionfor a massive field. Doing this directly in the continuum we get the standard continuumeuclidean equations

d∑j=1

∂Fij∂xj

−m2Diφ = 0 ,∂Diφ

∂xi= 0 (6.8)

6.2 Another amusing way to obtain the classical equation in theeconomic model

Here we derive the classical equations of motion directly in the original economic modelwithout going through the probabilistic interpretation. If we started with an arbitraryconfiguration of exchange rates and gold prices, we expect that speculators would startmoving around and earning money in the process. Let us focus on one of the banks thatsits between two neighboring countries. Let us say the currency of one is Pesos and theother is Dollars. If there are more speculators trying to buy Dollars than there are tryingto buy Pesos, then, in the real world, the bank would try to change the exchange rate sothat there is no imbalance.

In order to model this situation, we assume that the magnetic flux (6.2) or gold pricegradient (6.4) are both very small. We also assume that the exchange rates are very closeto one to one, and that gold prices are all very similar. In this world, the opportunities tospeculate are very small. We also assume that speculators follow only the two elementarycircuits in figures 16, 17. Of course, they can start from any country. We also make theimportant assumption the total amount of money carried by speculators following eachcircuit is proportional to the gain of each circuit

money carried by speculators = (constant)Fij or = (constant)Diϕ

This statement is a bit ambiguous because we did not specify the currency. However, wehave assumed that all exchange rates are close to one to one, therefore the units do notmatter for small values of these exchange rates. We also restrict the gauge transformationparameters ε(~n) to be small. When we say that the speculators carry an amount of moneyproportional to the flux, this money can be specified in any of the currencies on the circuit.It does not make a difference when we work to first order in the fluxes. As the speculatorsgo around the circuit, they will make a small profit proportional to the magnetic fieldFij(~n). As a consequence of our assumptions, this is small compared to the initial amount.In other words, it is a very small percentage.

In this situation the net amount of money flowing through a given bank, say the bankthat sits between the countries at point ~n and ~n + ~ei, is proportional to the number of

26

speculators crossing between these countries. Of course, speculators crossing from ~n + ~eito ~n count with a minus sign. We want this net flux of money to be zero so that the bankdoes not run out of either of the two currencies. Taking into account both the monetarycircuit and the gold circuit this imposes the condition

d∑j=1

(Fi,j(~n)− Fi,j(~n− ~ej))−Diϕ(~n) = 0 (6.9)

Note that all the elementary circuits that share the link going from ~n to ~n+ ~ei appear inthis sum.

Similarly we assume that the price of gold at each country adjusts so that there is nonet gold inflow or outflow. Otherwise the inhabitants of this country would change theirprice of gold. This implies

d∑i=1

Diϕ(~n)−Diϕ(~n− ~ei) = 0 (6.10)

This is summing over the contributions from speculators following the elementary Goldcircuits along all the bridges connected to a given country.

These equations, (6.10) and (6.9), become (6.8) in the continuum limit. Theseare the magnetic part of Maxwell’s equations in a time independent case. In this model,the Maxwell equations arise from the behavior of speculators that are present at shortdistances. In physics, there are theories where the equations of electromagnetism arisefrom the presence of a large number of very massive charged fields. At long distances theeffect of such particles is to induce the equations of electromagnetism.

6.3 Introducing time

So far, we have been discussing the model in Euclidean space, ignoring the time direction,or taking it to be equivalent to the space dimensions. Let us now include it more properly.We can think of one of the directions in our lattice as a time direction, say it is the dth

dimension, and label it by the index t. The exchange rate in the time direction is simplysaying that if you have some amount of money at some instant in time, then at the nextinstant your money will be converted to eAt(~n) times your original amount. Each instantin time has its own currency. Equivalently, you can think of At(~n) as the central bankinterest rate of the corresponding country. And you are required to deposit your moneyin the central bank. Of course there can be opportunities to speculate by going aroundcircuits that have one side along the time direction. You might say that it is impossibleto travel backwards in time. However, you can do the following. See figure 18. You makean arrangement with another speculator. You borrow some money and give it to him.Now you have debt and he has money. He stays in the original country and you move tothe neighboring country at the initial instant of time, you wait there till the next instant

27

(depositing the money in the corresponding central bank), and then you return to theoriginal country. If you did this properly, and if Fti is non-zero, he would end up withmore money that your debt. You can cancel the debt and share the profits with yourfriend. In physics, this would be analogous to a situation where you create an electron anpositron pair at some point in spacetime with some initial velocities so that they run awayfrom each other. Then the electric field pushes them back together at a later instant intime.

A (n)

A (n)

i

i

i

i

n+ e

n n+ ei

n+ e + ei

A (n + e )

A (n + e )t t

tt

t

t

Time

Electric Field

positron electron

(a)(b)

Space

Figure 18: (a) Economic model when we include the time direction. The vertical directionis time. Here green is money and red is debt. Following this circuit there can be gain ifFti is positive. (b) Corresponding process in real electromagnetism. Here some externalagent creates a positron and an electron moving in opposite directions. The electric fieldcurves their trajectories and makes them meet again. In this process there is a net “gain”.When particles try to take advantage of this gain, they end up moving as if they wereaccelerated by the electric field.

Let us now derive Maxwell’s equations. Let us consider the case with no gold, so thatspeculators can only carry money between different countries. We assume that we startwith a spatial lattice as before. For our spacetime we would start from a three dimensionallattice of countries. Let us take time to be continuous, and think of At as the central bankexchange rate. For simplicity, let us choose the currency of each country so that we canset At = 0. This is like making a continuous adjustment of the currency units. As before,we assume that the amount of money that speculators carry per unit time around theelementary spatial circuits is proportional to the magnetic flux of the spatial circuit. Ifthe net flux of money at a bank is non-zero, then the bank starts accumulating one of thetwo currencies. It will have an imbalance. The imbalance at the particular bank sitting

28

between the countries at ~n and ~n+ ~ei is changing as

dIi(~n)

dt= −

d−1∑j=1

[Fi,j(~n)− Fi,j(~n− ~ej)] (6.11)

Here Ii(~n) is the total imbalance of the bank. It is the difference between the amountof currency of the country at ~n + ~ei minus the total amount currency of the country at~n that the bank has. Now we add a new rule. We assume that when the bank sees animbalance Ii(~n) it starts changing the exchange rate with a speed which is proportional tothe imbalance

dAi(~n)

dt= Ii(~n) (6.12)

By taking a time derivative of (6.12) and using (6.11) we obtain

d2Ai(~n)

dt2= −

d−1∑j=1

[Fi,j(~n)− Fi,j(~n− ~ej)] (6.13)

which becomes of the Maxwell’s equations in the continuum limit. This is a wave equationwhich predicts the electromagnetic waves of figure 13. The other equation, which is Gauss’slaw, says ∑

i

dAi(~n)

dt− dAi(~n− ~ei)

dt= 0 (6.14)

This can be derived by assuming that there are speculators going around the time circuits,see figure (18). These speculators also carry an amount of money proportional to the gainon the circuit. The gain is proportional to dAi

dt. Demanding that there is no net amount of

money deposited at each of the countries central banks imposes (6.14). We can restore a

generic value of At by replacing dAi(~n)dt→ Fti = dAi(~n)

dt− [At(~n + ~ei) − At(~n)] in the above

equations.With similar assumptions we get the equation for a massive field when we include

gold. With gold one needs to assume that the price of gold obeys an equation of the formdp(~n)dt

= −G(n) where G(n) is the amount of gold at each country, etc.

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